There are several prior art techniques for measuring the distribution of the particle size in a sample by using light scattering. Generally, to measure the sizes of individual particles, for example, in a flowing stream of a liquid or gas, the particle-containing sample is illuminated by a constant light source and the intensity of light scattered by the particle is detected. A particle scatters the light by an amount directly related to the particle size; in general, bigger particles scatter more light than smaller particles. The relation between the amount of scattering and particle size may be determined either from theoretical calculations or through calibration process. With knowledge of this relation, for a single particle at a time, the detected scattered light intensity provides a direct measure of the particle size. The distribution of particle sizes in a sample may be determined by individually passing each particle in the sample, or a suitable portion of the sample, through the scattered light detection apparatus and tabulating the sizes of the various particles. In practice, this method is generally restricted to particles greater than 0.5 microns. Moreover, this method is relatively slow since particles must be detected individually. This technique is referred to in the prior art as optical particle counting (OPC).
A second prior art technique of particle sizing by light scattering is referred to as static or "classical" light scattering (CLS). This method is based upon illumination of a sample containing the particles-to-be-sized followed by the measurement of the intensity of scattered light at several predetermined angles. Because of intra-particle destructive interference, the intensity of light scattered from a particle depends on both the size and composition of the particle and the angle at which the measurement is made. This method of particle sizing based on the angular dependence of the scattered intensity can be used to measure the size distribution of a group of particles, as opposed to the first method noted above which is restricted to individual particles.
To implement the CLS measurement method, a sample of particles dispersed in a fluid is illuminated along an input axis, and the intensity of scattered light is measured at several predetermined angles. The scattered light intensity at each angle may be measured simultaneously with a multitude of detectors or consecutively, by moving a single detector around the sample to permit measurement of the intensity at each desired angle.
For large particles, for example, having diameters greater than 1 micron, the scattered light flux is concentrated in the forward direction relative to the input axis. Instruments for sizing large particles are referred to as laser diffraction devices. For sizing of smaller particles, for example, having diameters as low as 0.2 microns, the scattered light flux has significant magnitude both at lower and higher scattering angles relative to the input axis. The angular intensity measurements used on smaller particles are termed total integrated, or average, intensity measurements and may be displayed in a form known as Zimm plots.
A third prior art technique for particle sizing by light scattering is dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS) or quasi-elastic light scattering (QELS). See B. E. Dahneke, "Measurement of Suspended Particles by Quasi-Elastic Light Scattering," John Wiley & Sons, Inc., New York, 1983. This technique is based on measuring the time-fluctuations of the intensity of light scattered from an illuminated sample containing a group of particles which are diffusing through a fluid, that is, randomly moving due to collisions with solvent molecules and other particles. For example, the particles may be macromolecules dissolved in a liquid, where the macromolecules may be ionized by the loss of a small number of charged atoms.
In accordance with the DLS technique, scattered light intensity is measured as a function of time at a selected angle with respect to an illumination input axis. The light intensity detected at any instant at the detector is dependent on the interference between the light scattered from each illuminated particle in the scattering volume. As the particles randomly diffuse through the solution, the interference of the light scattered from them changes and the intensity at the detector therefore fluctuates. Since smaller particles diffuse faster, the fluctuations resulting from the motion of relatively small particles vary faster then those resulting from the motion of larger particles. Thus, by measuring the time variation of the scattered light fluctuations at the detector, information representative of the distribution of particle sizes is available. More particularly, the autocorrelation function of the measured intensity is related to the distribution of particle sizes in the fluid. Conventional DLS instruments such as the Model N4 photon correlation spectrometer manufactured by Coulter Electronics, Inc., Hialeah, Fla., provide autocorrelation signals for the detected intensity suitable for measuring distributions including particle sizes as low as 0.003 microns. Accordingly, such devices have a size measuring range extending considerably below the above-noted individual particle and CLS methods.
Particle sizing measurements by the known DLS techniques are generally made in the following manner. The particles-to-be-sized are suspended or dissolved in a fluid, forming a sample. The sample is illuminated by a laser beam directed along an input axis. Although a laser is generally used to generate the beam, a non-coherent light source may alternatively be used.
The light scattered from the particles in the sample is detected by a photodetector, such as a photomultiplier, which is positioned at a predetermined angle. The particular angle may be selected by the operator, but usually only one angle is measured at a time. The photodetector produces a signal which varies with time as the scattered light intensity incident on the photodetector varies. This time-varying signal is applied to an autocorrelator analyzer, to compute the autocorrelation function of the photodetector signal. Typically, the autocorrelator computes the value of the autocorrelation function of the detected scattered light at as many as one hundred discrete time points. This autocorrelation function contains the information about the fluctuations in the detected scattered light, from which information about the distribution of particle sizes in the sample can be extracted. Thus, the autocorrelation function (acf) is the raw data of a DLS measurement. While most conventional DLS measurements are performed using this autocorrelation step, it is known that the acf of the intensity signal corresponds to the Fourier Transform of the power spectrum of that signal. Accordingly, a spectrum analyzer may be used in place of the autocorrelator to generate a power spectrum signal including the same information representative of the particle size distribution as is resident in the autocorrelation function. The frequency domain information resident in the power spectrum signal can be used to determine the particle size distribution.
In the prior art, there are several techniques for extracting the particle size distribution from the acf. For use with these techniques, the relationship between the acf and the size distribution can be expressed as: EQU g(t)=K(x(r)) (1)
where g(t) is the acf (or a function closely related to the acf), x(r) is the sought distribution of particle sizes (x is a function of r, the particle radius), and K is a function (or operator, linear or non-linear) which relates particle size to the acf. Thus, given the exact form of K, the autocorrelation function resulting from any distribution, x(r), of particle sizes would be known.
Since the acf, g(t), is what is actually measured in practice, the above relation must be inverted to yield the particle size distribution: EQU x(r)=K.sup.-1 (g(t)) (2)
Accordingly, for the measured acf for a sample of particles, the size distribution for those particles can be extracted by applying the operator K.sup.-1 to the measured acf, g(t). The operator K.sup.-1 is the generalized inverse of the operator K. In practice however, the acf is "ill-conditioned" so that the inversion process is generally difficult and complex, although there are a number of known techniques for performing the inversion.
An example of the form K.sup.-1 for one particular commonly used prior art extraction technique is: EQU x=(K.sup.t K+.alpha.H).sup.-1 K.sup.t g (3)
In this example, x is a vector whose components are the proportions of the particles of each size, g is a vector whose components are the values of the acf at different points in time, as computed by the autocorrelator, K is a matrix relating x to g, and H is a matrix which increases the conditioning of the inversion. K.sup.t is the transpose of the matrix K. Alpha (.alpha.) controls the amount of conditioning imposed on the solution. The inverse operator K.sup.-1 in this case can be written K.sup.-1 =(K.sup.t K+.alpha.H).sup.-1 K.sup.t, where alpha (.alpha.) is a smoothing parameter determined conventionally. The inversion is usually performed along with some non-negativity constraints imposed on the solution; these constraints are formally part of the inverse operator K.sup.-1. Other known methods for inversion are the histogram method, the singular value decomposition method, the delta function method, and the cubic spline method.
The size distribution, x(r), obtained from this extraction or "inversion" process can be expressed either as a continuous distribution as implied by x(r), where the distribution is defined for particles of any size, or as a discrete size histogram expressed by the vector x, where the distribution of particle sizes is defined at only a set number of particle sizes. The vector x is representative of a group of number (x(r.sub.1), x(r.sub.2), . . . , x(r.sub.n)) giving the relative proportion of scattered light intensity from particles of size r.sub.1, r.sub.2, . . . , r.sub.n, respectively. The size distribution x(r) is referred to as a size histogram x herein below.
The size distribution x(r) and size histogram, x, are "intensity weighted" functions since these are representative of the relative proportion of particles as characterized by the relative amount of scattering intensity from particles of each size. However, these intensity weighted functions are dependent on the angle at which the measurement of scattered light was made. That is, the apparent proportion of particles of each size, as evidenced by the scattered light intensity contribution of particles of different sizes, depends on the angle at which the measurement is made. Thus, size distributions made at different angles cannot be directly compared using the intensity weighted distribution x(r) or histogram x.
Accordingly, if the amount of light scattered per particle, as a function of the scattering angle, is known, either through theoretical calculations or by an empirical method, the intensity weighted size distributions x(r) and histogram x at each angle can be directly compared by first converting those functions to corresponding mass, volume, or number weighted size distributions. For example, the intensity weighted histogram x may be converted in accordance with: EQU v=Cx
In this expression, v is the mass, volume or number weighted size histogram and C is the conversion matrix between the intensity weighted histogram, x, and the mass, volume or number weighted histogram, v. Similarly, the size distribution x(r) may be converted into a corresponding mass, volume or number distribution function v(r). Since all of these converted histograms and distribution functions provide the desired angle-independent information about the size distribution particles, they are referred to generally below as v and v(r), respectively.
A volume weighted histogram and distribution function provide a measure of the proportion of the total volume of particles in a sample as a function of particle size. For example, 50% of the volume of a sample of particles might come from particles of size 0.1 micron and the remaining 50% from particles of size 0.3 microns. Similarly, the mass weighted histograms and distributions provide a measure of the mass of particles in a sample as a function of size and the number weighted histograms and distribution of the numbers of particles in a sample as a function of size. For particles of the same density, the mass and volume weighted histograms and functions are the same. Volume, mass and number weighted size histograms and distributions are generally more useful than the corresponding intensity weighted size histograms or distributions since the former relate to quantities which can be directly measured by other means.
All of the prior art light scattering measurement techniques are characterized by low resolution and poor reproducibility, the principal drawbacks of such methods. With respect to DLS sizing measurements, efforts have been made to try to increase the resolution. The general methods used to increase resolution either attempt to improve the signal-to-noise ratio of the measurement by collecting intensity data over a long period or over a large number of short periods and then averaging the results, or by using intensity data collected at several angles.
With the latter method, the data collected at different angles are substantially independent, and therefore data collected at one angle complement those collected at other angles. For example, data collected at lower scattering angles are generally more sensitive to the presence of large particles in the sample while, conversely, data collected at large scattering angles are more sensitive to the presence of smaller particles. A sample containing both large and small particles can therefore be accurately sized by using the data from two or more angles, where relatively lower angle or angles provide information about the larger particles and relatively high angle or angles provide information about the smaller particles. In contrast, measurement at a single low angle would provide relatively little and possibly obscured information about the smaller particles and hence the sizing resolution would be poor.
The prior art method of using several angles to enhance the sizing resolution involves simply making measurements at two or more angles and averaging the volume weighted histograms resulting from the measurements made at the two or more angles. Symbolically, the process of combining information obtained at several scattering angles by averaging results can be expressed by: ##EQU1## where the subscripts 1, 2, . . . m refer to measurements made at the scattering angles .theta..sub.1 through .theta..sub.m. The inclusion of .theta. as an argument of the operator K.sup.-1 indicates that the inversion process, that is, the operator K.sup.-1, depends on the scattering angles. Each of the m intensity weighted histograms, x.sub.1, . . . , x.sub.m, may be converted to an angle-independent volume weighted histogram, v.sub.1 . . . v.sub.m, and then the m volume weighted histograms may be averaged to produce the "enhanced" resolution result, v: EQU v=(1/m)[v.sub.1 +v.sub.2 + . . . +v.sub.m ]
However, this volume weighted distribution, v, is not necessarily the solution which is the best fit to all the data. The size resolution obtainable for a single measurement at a single angle is quite low and the presence of particles of some sizes may not be detected at some angles. Thus, even when the intensity histograms are converted to volume histograms, the histograms obtained at different angles may give very different and apparently contradictory information.
Briefly, the invention disclosed in U.S. patent application Ser. No. 817,048 is an apparatus and method which provides a measure of the size distribution of particles dispersed in a fluid based upon an optimum combination of CLS measurements and DLS measurements, providing a resultant measurement characterized by relatively high resolution particle sizing. More particularly, in accordance with that invention, DLS data representative of the autocorrelation function, or power spectrum, of the detected intensity of scattered light at a plurality of angles about a sample, is optimally combined with CLS data representative of the average total detected intensity at those angles, to provide an angle-independent, high resolution size distribution v(r). The size distribution may be expressed in terms of the continuous function v(r) or the histogram v, and may represent distributions weighted by mass, volume, number, surface area, or other measures.
By way of example, in combining the DLS and CLS data, an angle independent volume weighted histogram may be determined from: EQU v=J.sup.-1 (g.sub.1 (t), g.sub.2 (t), . . . , g.sub.m (t); EQU i(.theta..sub.1), . . . , i(.theta..sub.m), i(.theta..sub.m+1), . . . i(.theta..sub.n))
where g.sub.1 (t), . . . , g.sub.m (t) are the determined autocorrelations of the detected light intensities at m scattering angles .theta..sub.1, . . . , .theta..sub.m, and where i(.theta..sub.1), . . . , i(.theta..sub.m),i(.theta..sub.m+1), . . . , i(.theta..sub.m+n) are the detected average intensities at the respective m scattering angles; .theta..sub.1, . . . , .theta..sub.m as well as n additional angles .theta..sub.m+1, . . . , .theta..sub.m+n, where m is an integer equal to or greater than one and n is an integer equal to or greater than zero. In this form of the invention, the DLS measurements are made at m angles and the CLS measurements are made at m+n angles, including the same angles at which DLS measurements are made. J.sup.-1 is a single operator which acts simultaneously on all of the autocorrelation functions and average intensity values to provide the "best fit" to all the data. This is in contrast to the m separate K.sup.-1 operators, one for each angle, set forth in equations (4) above. In various forms of the invention, rather than different angles, acf and intensity measurements may be made at the same angle, but under difficult conditions, for example, temperature, hydrodynamic solution characteristics, or polarization angles, which establish independent intensity characteristics at the sensor.
With the invention, the operator J.sup.-1 incorporates the information from the CLS measurements as well as the independent information from the DLS measurements, in a manner appropriately normalizing the autocorrelation functions measured at the different scattering conditions. For simplicity, the following descriptions will characterize the various measurements as being made at angles denoted .theta..sub.i although it is only necessary that the measurements be made under conditions which result in independent intensity characteristics.
The resultant distribution, v, based upon the J.sup.-1 transformation of the autocorrelation functions and the classical scattered intensities, simultaneously in a single procedure, provides an increase in the sizing resolution of the determined particle distributions compared to the prior art techniques which are based upon either the autocorrelation functions of the classical scattering intensities, but not both.
Briefly, according to the invention disclosed in U.S. patent application Ser. No. 817,048, a system is provided for measuring the size distribution v(r) of particles dispersed in a fluid sample, where r is representative of particle size. The system includes means for illuminating the sample with a light beam directed along an input axis. Either a coherent or a non-coherent light source may be used.
A light detector detects the intensity of light from the light beam at m points angularly dispersed from said input axis at a plurality of angles .theta..sub.1, . . . , .theta..sub.m, where m is an integer equal to or greater than one. The detector generates m intensity signals, each of the intensity signals being representative of the detected intensity of the light from the light beam as a function of time at a corresponding one of the m points. In various forms, the invention may be embodied in a homodyne or a heterodyne configuration. In the homodyne form, only scattered light is detected at the m points during the intensity signal measurements, while in the heterodyne form, a portion of the beam is directly incident on the detector at the m points, so that the intensity signal corresponds to a beat signal resulting from both scattered and non-scattered portions of the light beam.
In one form, an autocorrelation processor generates m correlation signals each of the correlation signals being representative of the autocorrelation function of a corresponding one of the intensity signals. Each of the correlation signals equals an associated transformation J.sub.i of the distribution v(r), where i=1, . . . , m. The transformations may be linear or non-linear. Since the autocorrelation functions for the intensity signals are the Fourier Transforms of the power spectra of those signals, the autocorrelation processor is, in one form of the invention, an autocorrelator which directly generates the m correlation signals as m time domain autocorrelation signals g.sub.i (t), where t is time and i=1, . . . , m. In other forms, the autocorrelation processor includes a spectrum analyzer which generates the m correlation signals as m frequency domain power spectrum signals G.sub.i (f), where f is frequency and i=1, . . . , m. Since the power spectrum signal is the Fourier Transform of the autocorrelation signal, the power spectrum signals G.sub.i (f) may be used to provide the same information as the autocorrelation signals g.sub.i (t).
A light detector also detects the time average intensity of scattered light from the light beam at the m points as well as n additional points angularly displaced from the input axis, where n is an integer greater than or equal to zero. The latter detector generates average signals representative of the time average of the intensity of scattered light detected at the respective m+n points.
A size processor, responsive to the correlation signals and the average signals, generates a signal representative of the distribution v(r). The size processor generates a composite correlation signal representative of a weighted direct sum of the m correlation signals. The size processor determines a composite transformation operator J.sup.-1 which is related to the transformations J.sub.i and the n average signals.
The size processor transforms the composite correlation signal in accordance with the determined composite transformation operator thereby providing a resultant signal which incorporates the size distribution information of both the CLS and DLS data and is representative of the size distribution v(r). In accordance with the invention, either the composite correlation signal or the composite transformation operator is substantially scaled to the average intensities of the scattered light at the respective ones of the m points. This scaling, or normalization, permits the DLS data represented by the composite correlation signal to be optimally combined with the CLS data represented by the average signals.
In one form of the invention, the transformations J.sub.i are linear transformations and the composite transformation operator J.sup.-1 is the generalized inverse of the operator corresponding to the direct sum of the operators for the associated transformations J.sub.i. The inverse transformation operator J.sup.-1 may correspond to the inverse of the matrix corresponding to the direct sum of the associated transformations J.sub.i. Alternatively, the operator J.sup.-1 may correspond to EQU [J.sup.t J+.alpha.H].sup.-1 J.sup.t
where J is the matrix corresponding to the direct sum of the matrices coresponding to the associated transformations J.sub.i, J.sup.t is the transpose of the matrix J, H is a conditioning matrix, and alpha (.alpha.) is a smoothing parameter. Further, all components of the vector representative of the distribution v(r) may be constrained to be greater than or equal to zero.
In another form, the associated transformations are non-linear, with the size distribution being characterized by v(r,p), where p is a characterization parameter vector having k components. In this form, the composite transformation operator J.sup.-1 is the p solution algorithm for ##EQU2## where i is an integer 1, . . . , m, j is an integer 1, . . . , q, l is an integer 1, . . . , k, p.sub.l is the l.sup.th component of p and where g.sub.i (t.sub.j) is the autocorrelation function of the intensity signal for the i.sup.th of said angle at the j.sup.th time interval and J.sub.ij is an operator related to the associated transformations. The model size distribution v(r,p) may for example be characterized in terms of parameters r and .rho., the mean particle size of the actual size distribution, and the standard deviation of the actual size distribution, respectively. The solution algorithm for the p.sub.l minimizes the squares of the residuals J.sub.ij [v(r,p)]-g.sub.i (t.sub.j) for the various points in time t.sub.j for the various acf's g.sub.i. More particularly, v(r,p) may have the form: ##EQU3## where the J.sub.ij operator has the form: where .GAMMA.(r,.theta..sub.i) has the form: ##EQU4## where n is the refractive index of the sample, .lambda. is the wavelength of the light illuminating the sample, k.sub.B is Boltzmann's constant, .eta. is the viscosity of the sample and T is absolute temperature.
In another form, the composite correlation signal operator controls the weighted direct sum of the m correlation signals to be unity normalized and the composite transformation operator is substantially scaled to the average intensities of light scattered from the light beam at the respective ones of the m points. In yet another form, the composite correlation signal generator controls the weighted direct sum of the m correlation signals to be substantially scaled to the average intensities of light scattered from the light beam at the respective ones of the m points.
In other forms of that invention, the general method of using information exacted from measurements at two or more scattering angles can be applied to determine size and shape information about particles which are rod-like, ellipsoids or other forms, including "Gaussian coils".
In other forms of that invention, instead of making CLS and DLS measurements at two or more scattering angles, such measurements may be made at one angle under different sets of conditions, for example, different temperatures or hydrodynamic solution characteristics or polarization of the light beam, providing complementary information which is processed to yield enhanced particle characteristic resolution for a wide variety of dynamic systems.
A critical aspect of the invention disclosed in U.S. patent application Ser. No. 817,048 is the recognition that by combining CLS (classical light scattering) data and DLS (dynamic light scattering) data from multiple scattering angles, the resolution and repeatability of particle sizing may be improved over prior art CLS or single angle DLS measurements.
To combine CLS data and DLS data is difficult and the determination of conditions under which the CLS and DLS data can be combined in a single, simultaneous analysis is a complicated process. In particular, one major problem in combining these two types of light scattering measurements is in properly normalizing the DLS data taken at different angles. Briefly, "normalization" refers to adjusting the amplitudes of the autocorrelation functions (acfs) at different angles with respect to the CLS data so that data from all the DLS angles can be analyzed within one comprehensive model.
The normalization of the DLS data to the CLS data is important in view of the following. A CLS measurement at a single angle is the average value of the light scattered from particles of all sizes in the sample being measured, weighted by the intensity of light scattered by particles of each size. The amount of light scattered at a particular angle from a particle of a particular size depends on both the size of the particle and the scattering angle. To make particle size measurements using CLS data alone (that is, average intensity measurements at a multitude of angles), measurements at the selected angles are made and then an analysis procedure is used to find a particle size distribution corresponding to the measured pattern of angular scattering intensity. The selected distribution must be such that the light scattered from every size of particle in the distribution, weighted by the angle dependent intensity of light scattered by particles of each size, must be close, at each angle, to the measured value of the average scattered intensity at that angle.
CLS data is thus a subset of DLS data since the magnitude of the CLS intensity at a particular angle is simply the value of the zero time point of the homodyne DLS autocorrelation function (acf) at that angle. The relation between CLS data and DLS data may be appreciated by considering a plot of the CLS data in a cartesian (X-Y-Z) coordinate system in which the x axis represents scattering angles (e.g. from 0 to 180 degrees), and the y axis represents scattering intensity. In such a coordinate system, a given distribution of particles would be characterized as a curve in the X-Y plane. The height of the curve at any point would be the scattering intensity at the angle corresponding to that point. This curve, or at least a number of points along this curve, correspond to a CLS measurement.
The z axis represents the delay time of an acf of the scattered light intensity. Generally, acf's have the form of decaying exponentials (or sums of decaying exponentials). A number of discrete points along the CLS curve (in the xy plane) correspond to the angles at which DLS measurements may be made. The DLS data for each point (or angle) defines a curve in a plane parallel to the Y-Z plane. These acfs (at one or more angles) are the DLS data. The shape of the acf at each angle in general would be slightly different unless all the particles were the same size. Thus, the matrix of acf data is used in a measurement of the type defined in U.S. patent application Ser. No. 817,048. In the prior art, only the curve in the X-Y plane alone or a single one of the acfs would be analyzed to obtain a (low resolution) particle size distribution. The system of U.S. patent application Ser. No. 817,048 provides particle distribution measurements utilizing the two dimensional "measurement" surface on which all the values of the acfs at all delay times and all scattering angles lie. The intersection of this surface with the X-Y plane is the CLS data curve.
To analyze the matrix of data shown, a model is made describing how the measurement surface, or more precisely, the discrete points on the measurement surface at which actual measurements are made, varies when the parameters (e.g. histogram bin heights) of the particle size distribution vary. Then, when a particular measurement is made, the particle size distribution which leads to a measurement surface closest to the measured measurement surface is the best estimate of the true particle distribution. The best estimate is found by a curve fitting algorithm, such as a non-negative least square (NNLS) algorithm.
The importance of normalization to this process is that in order to accomplish the curve fitting, the measured acfs (which are ordinarily measured without regard to their absolute height) need to be normalized so that their amplitudes, i.e. zero time values, are exactly the amplitudes of the CLS data curve at the angles at which the acfs are measured. If this normalization is not undertaken, the measured measurement surface generally differs from the model measurement surface since in the model measurement surfaces all acfs at time zero must lie on the CLS data curve. Thus, the measured data are normalized to be appropriate to this model.
In order to perform the DLS data normalization, CLS data (i.e. the average intensity of scattered light) must be measured at every DLS angle to obtain the normalization constants. For this reason, U.S. patent application Ser. No. 817,048 defines an analysis in which the DLS data is measured at m angles (m&gt;0) and the CLS data is measured at those m angles and at n additional angles, i.e. CLS data needs to be measured at all the DLS angles and optionally at some additional angles.
However, it is difficult to accurately obtain the CLS data used for the DLS normalization. More importantly, small errors in the normalization constants lead to serious errors in the extracted particle size distribution. Therefore, it is desireable to find an improved model by which to analyze the data.
Because of the mathematical form of acfs, the amplitude information is mathematically separable from the rest of the information contained in the acf, namely the information contained in its shape. By performing a mathematical transformation in one of several ways, the DLS amplitudes can be removed from the model, thereby allowing the curve fitting to be accomplished without knowledge of the amplitudes of the acfs, i.e. without a normalization in the manner defined in U.S. patent application Ser. No. 817,048.
It is an object of the present invention to provide an improved apparatus and method for measuring the distribution of particle sizes dispersed in a fluid.
Another object is to provide an improved particle size distribution measuring apparatus and method characterized by relatively high resolution.